MiS Preprint Repository

We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV ( that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint

Frobenius Statistical manifolds, Geometric invariants & Hidden symmetries

Noémie Combe, Philippe Combe and Hanna Nencka


In this paper, we explicitly prove that statistical manifolds have a Frobenius manifold structure. This latter object, at the interplay of beautiful interactions between topology and quantum field theory, raises natural questions, concerning the existence of Gromov–Witten invariants for statistical manifolds. We prove that an analog of Gromov–Witten invariants for statistical manifolds (GWS) exists, and that it plays an important role in the learning process. These new invariants have a geometric interpretation concerning intersection points of paraholomorphic curves. In addition, we unravel the hidden symmetries of statistical manifolds. It decomposes into a pair of totally geodesic submanifolds, containing a pair of flat connections. We prove that the pair of pseudo-Riemannian submanifolds are symmetric to each other with respect to Pierce mirror.

MSC Codes:
53B99, 62B10, 60D99, 53D45
statistical manifold, Frobenius manifold, Gromov–Witten invariants, Paracomplex geometry

Related publications

2021 Repository Open Access
Noémie Combe, Philippe Combe and Hanna Nencka

Frobenius statistical manifolds and geometric invariants

In: Geometric science of information : 5th international conference, GSI 2021, Paris, France, July 21-23, 2021, proceedings / Frank Nielsen... (eds.)
Cham : Springer, 2021. - pp. 565-573
(Lecture notes in computer science ; 12829)